BA CAI will run your itinerary through their London rate desk so I would count on the 20 segment rule being enforced. If your itinerary is simple, a couple of days in advance should be enough but I'd suggest at least a week to allow for contingencies.

While I agree that open tickets make life easier, they sometimes make ticketing tougher when dealing remotely. I had problems with trying to convince London that I was only making transits of HKG and LHR due to the lack of specific flights and dates. Finally got it ticketed but they ran me down to the wire.

BTW, BA CAI is only open Mon-Thur although Mrs. Sobhi is usually in the office on Friday.